Vortex patch with wall interaction as new benchmark test

Cent. Eur. J. Eng. • 1(4) • 2011 • 390-400DOI: 10.2478/s13531-011-0037-2

Central European Journal of Engineering

Vortex Patch with Wall Interaction as NewBenchmark Test

Research article

Ziemowit Milosz Malecha 12∗

1 Integrated Applied Mathematics Program, University of New Hampshire,Durham, USA

2 Faculty of Mechanical and Power Engineering, Wroclaw University of Technology,Wroclaw, Poland

Received 7 March 2011; accepted 24 June 2011

Abstract: In this paper, a new computational benchmark test for fluid dynamics is presented. The new benchmark is basedon the interaction of a single vortex structure (vortex patch) with a wall. It will be shown that it is possible todistinguish two critical or threshold values of the Reynolds number in the considered flow. The increase of theReynolds number causes the appearance of the vortex bubble in the near-wall region first, and then next, theeruption of the boundary layer phenomenon. Further increase of the Reynolds number causes the flow to bemore complex. The eruption phenomenon becomes more intense and also shows its regenerative nature.

Keywords: Benchmark test • Boundary layer • Rruption • Vortex-in-cell method© Versita sp. z o.o.

1. Introduction

Vorticity is fundamental in the mechanics of fluids. Eachreal flow has a non-zero vorticity. A great number of phe-nomena in hydrodynamics is analyzed from the perspectiveof vorticity dynamics. Many flows produce characteristicvortices, of which their structure and behavior can give usmuch important and interesting information about the con-sidered flow. Moreover, these vortex structures appear invery specific circumstances and act in very specific ways,depending on the Reynolds number. A well known exam-ple can be found in the flow around the cylinder, or in theflow in the rectangular box with one moving wall (cavityflow). The frequency of the vortices which have been shed∗E-mail: [email protected]

from the cylinder, and the appearance and shape of thecorner vortices in the cavity flow, change very specificallywith the Reynolds number. They often serve as benchmarktests for new computational algorithms or experimental se-tups [1].In an incompressible flow, vorticity can be generated onlyon the rigid wall. Vortex production on the wall is forcedby the fluid viscosity (no-slip condition). The productionof the vorticity on the wall can be interpreted as neces-sary for the maintenance of the no-slip condition [2, 3].Introduction of the vorticity from the wall can take placethrough short range diffusion, as in laminar flow, or canhappen abruptly through vorticity eruption from the walllayer [4, 5].In some authors’ recent works [4, 6], the numerical inves-tigation of the boundary layer eruption phenomenon waspresented. The research was mainly focused on the mu-tual interaction of the single vortex structure with the plain390

Z.M. Malecha

and rigid wall. However, the flow itself does not alwayslead to the eruption. Further studies have shown that thevortex – wall interaction has a different nature in eachdifferent flow regime (different Reynolds number). Thechange of the flow is characterized by the appearance ofnew and distinct flow patterns and phenomena. Becausethe changes in the flow take place for a specific value ofthe Reynolds number, it can be considered a benchmarktest as well.The eruption phenomenon is connected with the process,already described by Prandtl, as the separation of theboundary layer [7]. The process of separation takes placewhen two fluid particles with opposite velocity vectorsmeet at one point on the wall (Fig. 1).

Figure 1. Prandtl’s sketch of the streamlines and velocity profiles inthe vicinity of the separation point S. To the right of theseparation point, a reverse flow is established.

Denoting the velocity component along the wall by u, andperpendicular to it by v , at the separation point we have:∂yu = 0, at y = 0. (1)

Which means that the shear stress at the wall is zero.The condition above is important only for steady flowswith a low Reynolds number. It was demonstrated for thefirst time by Prandtl [7, 8].Moore, Rott and Sears (MRS) [9–11] presented argumentsindicating that the condition (1) is insufficient for unsteadyflows. According to MRS, the unsteady separation takesplace when the stagnation point and the point of zerostress are located within the flow. The new condition be-comes:∂yu = 0, and u = 0. (2)

In this paper, it will be shown that the two conditionsmentioned above will manifest as a result of the vortex–wall interactions for different Reynolds numbers. The nu-merical technique used in the work is the Vortex ParticleMethod [12]. The main aim of this work is to show that

the considered flow can serve as a new benchmark test.The emphasis will be put on the alternation of the flowstructure as a result of increasing the Reynolds number.The paper is organized as follows: first, the necessarymathematical background, and then a detailed descrip-tion of the Vortex Particle Method is presented. The nextsections show the numerical results, a discussion, and con-clusions.2. Mathematical background andVortex Particle MethodThe vorticity form of the two-dimensional Navier–Stokesequations of the viscous and incompressible fluid motionis:

∂tω+ u · ∇ω = ν∆ω, (3)∆ψ = −ω; u = ∂yψ, v = −∂xψ. (4)

where u = (u, v) is the velocity vector, ν is the kinematicviscosity coefficient, ∆ = ∂2x + ∂2

y is the Laplace operatorand ω = ∂xv − ∂yu.The equations (3) and (4) must be complemented by theboundary conditions for ω and ψ, and by the initial con-dition for ω.In vortex methods, the viscosity decomposition algorithmis commonly used [12]. This means that the equation (3)is solved in two steps: first, the inviscid equation:∂tω+ u · ∇ω = 0 (5)

then, the viscous diffusion equation (Stokes problem):∂tω = ν∆ω. (6)

In this paper, equations (5) and (6) were solved by VortexParticle Method.2.1. Vortex Particle MethodFrom the equation (5), we can see that the vorticity re-mains constant along the trajectories of the fluid materialparticles (dω/dt = 0). Mathematically, this fact can beexpressed as:

ω(x(t, α), t) = ω(α, 0)where x(t, α) denotes the position of the particle at timet, which at the initial instant t = 0, was in the positionα . According to the third Helmholtz theorem [13], vorticity

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Vortex Patch with Wall Interaction as New Benchmark Test

lines move with the ideal fluid. This means that the vor-ticity field evolution can be expressed as the movement ofthe infinite set of the vortex particles:dxpdt = u(xp, t) x(0, α) = α, (7)

where α = (α1, α2) means Lagrange coordinates of fluidparticles.By solving the Poisson equation for the stream function(4) and differentiating, the velocity field can be presentedas dependent on the vorticity distribution:u(x) = ∫ K(x− x′)ω(x′, t)dx′ (8)

where:K(x) = 12π | x | (−y, x), and | x |=√x2 + y2.

The equation (8) is a fundamental formula for direct vortic-ity methods [14]. The direct numerical algorithm to solveequation (8) has square complexity, which is exceedinglylarge. One of the approaches used to avoid this, is calcu-lation of the velocity, which is done by solving the Pois-son equation for the stream function (4) on the numericalmesh. Subsequently, the velocity from the mesh nodes isinterpolated onto particle’s positions. Such an approachsignificantly accelerates calculations, and for this reason,it was used in this work [12].For the numerical calculations, the infinite set of differ-ential ordinary equations (7) must be replaced with thefinite set. In order to achieve this, the space of La-grange variables is covered by a regular mesh (i∆x, j∆y),(i, j = 1, . . . , N), ∆x = ∆y = h. A similar mesh is alsoused to solve the Poisson equation, for the stream function,by the finite difference method. Next, the initial vorticityfield is replaced by a distribution of vortex particles. Cir-culation is assigned to each particle:Γp(xp) = ∫

Ap

ω (x, y) dA ≈ h2ω (9)where Ap = h2 denotes cell area with index p = (i, j),whereas ω is the mean vorticity value for the cell. TheVorticity field is approximated by the sum of Dirac discretemeasures:

ω(x) ≈ N∑i=1 Γδ(x− xp) (10)

where N is the number of particles and δ denotes theDirac function. The solution of the equation (5) in the time

interval (tn, tn+1) is obtained by solving a set of differentialequations:dxpdt = u(xnp(t), t), xp(tn) = xnp, tn ≤ t ≤ tn+1, p = 1 . . . N(11)and the new positions of the particles are approximatesolutions of the equation (5), for the time t = tn+1:

ωn+1(x) = N∑p=1 Γpδ(x− xn+1

p ), xn+1p = xp(tn+1) (12)

Circulations of particles change with time because of dif-fusion. This process is modeled by solving the diffusionequation (6). In the present work, this equation was solvedby the particle strength exchange method (PSE). The mainidea of the PSE method is to replace the differentialLaplace operator with the integral operator ∆ε , [12]:∆ε(x) ∼= 1

ε2∫ (ω(y)− ω(x))ηε(y− x)dy (13)

The function ηε = ε−2(x/ε), (ε > 0) is a symmetric cut-off function, satisfying specific conditions for its moments.For the PSE method to be effective, the relation ε/h ≥ 1where h is the cell size, must be satisfied, i.e. the supportsof adjacent particles must overlap. Following equations(12) and (13), the change of the particle intensity due toviscosity can be expressed as:dΓpdt = νε−2∑

q

(Γq − Γp) η(xp − xqε

). (14)

The function η(x) can have infinite support, but then it re-quires calculation of the interaction of each vortex particlewith all other particles in the flow domain. This meanssquare computational complexity, proportional to O(N2),where N denotes the number of vortex particles. A solu-tion of this problem is to apply the function η(x) with finitesupport. In this paper, a second order function with finitesupport was chosen:η(x) = { C1+|x|2 for |x| ≤ 20, for |x| > 2 (15)

where C = 0.835 was calculated to satisfy the conditionfor the second moment: (∫ x2η(x)dx = 2) [12]. In the caseof function with finite support, computational effort is pro-portional to O(mN), where m refers to the number of par-ticles within the support of the cut-off function. Becausem << N and m are constant, the computational effort

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is proportional to O(N). In the algorithm developed forthis work, the kernel (support) of the function (15) coveredexactly 9 neighboring vortex particles (m = 9).Vortex particle methods belong to the class of self-adaptive methods. Vortex particles tend to gather in theregions of high velocity gradients. This can be deleteriousto the precision of the viscous effect modelling, which cansignificantly alter the velocity field. In order to avoid theseproblems, it is necessary to apply an additional procedureensuring a uniform distribution of vortex particles in theflow domain. This process is called remeshing. When vor-tex particles are uniformly distributed, the cut-off function(15) always covers exactly the same number of particles.This can be achieved when remeshing is performed in eachtime step. In this work, such an approach was adopted.After the advection step, vortices were distributed on aregular grid coinciding with the numerical mesh used tosolve the Poisson equation (4). Reorganization of the par-ticles’ location and circulation was done by the secondorder interpolation kernel:φ(x) =

1− x2, 0 ≤ x < 12 ,(1− x)(2− x)/2, 12 ≤ x < 32 ,0, x ≥ 32

(16)For particles located in the vicinity of the boundary (closerto the wall than 1/2h), the interpolation kernel was [15](Fig. 2):φ(x) =

1− 32x + 12x2, for nodes 1,2,3,x(2− x), for nodes 4,5,6,x(x − 1), for nodes 7,8,9,0, for all other nodes, (17)

The two-dimensional interpolation kernel is a product ofone-dimensional kernels:φ∗(x, y) = φ(x)φ(y). Finally, theinterpolation proceeds as follows:Γ∗i (x∗i , y∗i ) ≈ N∑

j=1 Γj (xj , yj )φ(x∗i − xj )φ(y∗i − yj ) (18)where Γ∗ denotes new circulation of the vortex particle inits new position (x∗i , y∗i ).To advance the position of the vortex particles in time,the set of differential equations (11) was solved by thesecond order Euler-improved method. The velocity of theparticles between the grid nodes was calculated using theinterpolation formula:

u(xp) =∑j

uj lh(xp − xj ) (19)where lh is a basic bilinear Lagrange function.

Figure 2. Remeshing of vortex particles close to the wall.

2.2. Vorticity Boundary Condition.Describing the fluid motion using vorticity and streamfunction simplified the equations of the motion, but com-plicated the realization of the no-slip and no-penetrationcondition at the wall. Both normal and tangent compo-nents of the velocity field should vanish. The constantvalue boundary condition for the stream function at thewall ensures the zero value of the normal velocity compo-nent. However, the realization of the no-slip condition isnot as straightforward.In numerical practice of the vortex methods, realization ofthe no-slip condition is achieved by generating the appro-priate amount of vorticity on the rigid wall. The vorticitycan be generated by applying the proper Dirichlet or Neu-mann boundary conditions [16, 17]. In this work, a secondtechnique was adopted [18]. Using this approach, the wallis treated as a vorticity layer with intensity γ = [uω]. Inthis instance, uω represents the velocity induced on thewall by the vorticity located inside the flow domain. Thesquare bracket denotes a jump in the velocity value fromzero to uω. The no-slip condition is achieved by the elim-ination of this undesirable vorticity layer by assigning aspecific value of the vorticity flux (∂yω) at the wall:

∂yω = − γν∆t , (20)

where the y coordinate is perpendicular to the wall, andγ refers to the intensity of the vorticity layer along thewall.To introduce a new vorticity, generated by the vorticityflux, into the flow domain, an additional initial-boundaryvalue problem for the diffusion equation must be solved

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[16]:∂tω = ν∆ω,ω(x, y, t = 0) = 0,∂yω = − γ

ν∆t .(21)

It should be noted that the initial condition for the prob-lem (21) is equal to zero. In the present work above, theequation was solved using a finite difference method. Sub-sequently, the vorticity obtained by solving the equation(21) was added to the vorticity already existing in the fluidand converted to particles.3. Numerical Calculations and Re-sultsThe flow of our interest is the flow induced by a two-dimensional vortex (vortex patch) moving along the wall.Figure 3 shows the schematics of the computation domaintogether with a vortex patch. The distance between thevortex center and the wall is d, while r is the vortex ra-dius. To capture the characteristic features of the flow as afunction of the Reynolds number, the vorticity equation (3)was transformed to a non-dimensional form. Characteristicquantities were defined as follows: length L = d, velocityU = Γ/2πr, and time T = L/U . Using non-dimensionalvariables x′ = x/L, t′ = t/T , and u′ = u/U , equation (3)takes the form:

∂tω+ u∂xω+ v∂yω = (1/Re)∆ω, (22)where Re = UL

ν = Γ2πν · dr is a Reynolds number, andΓ = ω0πr2, where ω0 is a initial vorticity of the patch.The size of the computational domain was chosen throughtrial and error to minimize the influence of the boundarycondition on the numerical results. Finally, the figuresL = 8 and H = 7 were chosen for computational domainin non–dimensional units. To solve the Poisson equationfor the stream function (4), the boundary conditions usedwere, as follows: a periodical boundary condition in thedirection of x, and the Dirichlet condition ψ = 0 for thebottom (y = 0) and top (y = H) boundary. The initialposition of the vortex patch was x0 = 7.5, y0 = d = 1.Its initial radius was r = 0.3 and its vorticity was ω0 =−1.25. The mesh size and the time step were ∆x = ∆y =0.01 and ∆t = 0.01, respectively.One time step, from tn to tn +∆t, of the presented numer-ical method, proceeded as follows :

1. Redistribution of the particles’ circulations onto themesh nodes (16), (17).

Figure 3. Computational domain of the considered problem with vor-tex patch in its initial position.

2. Calculation of the stream function (4) by the finitedifference method (using a fast, elliptic equationsolver).3. Calculation of the velocity on the numerical mesh(using the second order formula).4. Determination of the vorticity layer γ = us andsolution of the diffusion equation (21).5. Displacement of the vortex particles, according to(11).6. Remeshing of the vortex particles onto a regulargrid ( formula (16), (17)).7. Changing of the particles’ circulations, as a resultof the diffusion, using the PSE method (14).

The diffusion equation (21) was calculated by the fast,elliptic equation solver as well. For this purpose, thetime derivative was replaced by its finite difference form:∂tω|t=tn+1 ≈ (ωn+1/∆t), (ωn = 0) , which resulted in theelliptic equation:

∆ωn+1 − ωn+1ν∆t = 0,

∂yω|y=0 = − usν∆t , ω|x=0 = ω|x=L, ω|y=H = 0 (23)

3.1. Results and DiscussionA vortex patch, characterized by negative vorticity, in thepresence of the wall beneath it, moves along it from rightto left. In analyzing the velocity field induced by the patch,we can observe that the maximum velocity is directly belowit. The pressure in this place has minimal value. To theleft from the vortex structure, in the direction of motion,the velocity decreases while the pressure increases. Thismeans that the direction of the pressure gradient vector(its horizontal component) is opposite to the horizontal

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Figure 4. Interaction of the vortex patch and the wall for Re = 500. A strong diffusion process can be observed. The figure shows the vorticityfield with selected streamlines.

Figure 5. Interaction of the vortex patch and the wall for RekrI = 1000. Near the wall, a new vortex structure is created, t = 10. The Figurepresents the vorticity field with selected streamlines.

component of the velocity field. Such a pressure gradientslows down the fluid motion, and especially affects theflow behavior close to the wall. For certain values of theReynolds numbers, this can provoke the appearance ofthe near-wall recirculation zones, or the boundary layereruption phenomena.By increasing the Reynolds number, we can observe dis-tinguishably different patterns in the flow field, which cansuggest a change in the flow regime. In performing care-ful computations, we can define two critical values of theReynolds numbers, which announce the appearance of thenew phenomenon.Figure 4 shows the considered flow for Reynolds numberRe = 500. In the figure, the vorticity field, along with

the selected streamlines induced by the vortex patch, ispresented. In this case, the flow is under strong influenceof diffusion and vorticity, just decaying with time.The flow stays qualitatively the same, till the Reynoldsnumber reaches its first critical value: RecrI = 1000,Fig. 5. The viscous forces are not as dominant as before,and we can observe the appearance of the recirculationzone near the wall (bubble vortex). The bubble developsup to a certain size and then it ceases because of theviscosity.Further rising of the Re number does not change the flowsignificantly. The only visible feature is the stretching ofthe vortex bubble in the direction normal to the wall, whatcauses the change of its shape. The situation changes395


Figure 6. Interaction of the vortex patch and the wall for RecrII = 5000. The eruption phenomenon takes place in the flow, which predicts a newflow regime. For time t = 15, a new vortex structure, detached from the wall, is visible. Inside it, the fluid elements from the near-wallregion are enclosed.

drastically after the Reynolds number reaches its secondcritical value: RecrII = 5000, Fig. 6.For RecrII = 5000, a new significant phenomenon mani-fests itself in the flow. It can be observed that the near-wall recirculation zone is intensely stretched in the direc-tion normal to the wall, which finally results in splitting.Streamlines form the characteristic ”figure of 8” shape witha saddle point (frame t = 11.25 in Fig. 6). This processresults in the eruption phenomenon, which manifests itselfas an ejection of a portion of the fluid from the boundarylayer to the outer flow. This portion is enclosed in thenew vortex structure (closed streamlines) (frame t = 15 inFig. 6).Increasing the Reynolds number above the RecrII causesmore intense and frequent eruptions, which make the flowto be more and more complex. Figures 7 and 8 present theconsidered flow for Re = 10000 and Re = 50000, respec-tively. In this case, the generated vortex structure on thewall has high intensity and does not disperse, but goesaround the primary vortex patch. This creates successive

vortex bubbles and eruptions (frames t = 28, t = 56 inthe Fig. 7)For Re = 50000, not only one vortex structure but a se-quence of vortex structures, detach from the wall (Fig. 8,frame t = 85). For such high values of the Reynolds num-ber, eruptions of the boundary layer causes highly complexflow and can give some ideas on near-wall turbulence.4. ConclusionsThe presented results show that the flow, induced by themotion of the vortex patch above the wall, can serve asa benchmark test for new computational algorithms orexperimental setups. Numerical results indicate that itis possible to distinguish two critical Reynolds numbers:RecrI = 1000 and RecrII = 5000 in the considered flow.Exceeding one of these threshold values causes the flowto change qualitatively. After crossing the threshold valueof the RecrI , we can observe the appearance of the near-

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Z.M. Malecha

wall recirculation zone – vortex bubble. This critical valueindicates that viscous forces are not predominant in theflow any more. Inertial forces start to be important aswell. In between the Reynolds number RecrI = 1000 andRecrII = 5000, the flow preserves its basic structure, onlythe shape of the vortex bubble becomes more stretched inthe direction normal to the wall.Next, and more significantly, a change in the characterof the flow occurs when the Reynolds number crosses thesecond threshold value. Above RecrII = 5000, the erup-tion process takes place, which drastically changes thedynamics of the flow. The vortex bubble splits apart anda secondary vortex patch enters the flow domain. Thenew vortex structure alternates the motion of the primaryvortex patch.The increase of the Reynolds number above the secondcritical value causes the flow to become more complicated.It is possible to observe the regenerative nature of theeruption phenomenon: the creation of successive vortexbubbles and eruptions. In these complications, there ap-pears a cascade of vortices that detach from the wall andmove around the primary vortex patch, evoking the se-quence of secondary eruption events.Results from this paper can be especially used to testother computational methods which focus on vortex-wallinteractions. However, they can also be used as pre-dictions of significant alternations in any flow with rigidboundaries. Being close to the critical value of Re can in-dicate that a transition to a different flow regime is likelyto occur. Since the numerical value of critical Re dependson its definition, that is why this should be considered inapplication to the different types of flow.Moreover, the presented results can also have piraticalmeaning. The observed phenomena can be seen as typicalphenomena for flows with rigid boundaries. The value ofthe Reynolds number can tell us what kind of phenomenonis likely to occur. From the perspective of specific appli-cations, in some cases it is more desirable to be belowor above certain critical values of the Reynolds number.For example, in the flows connected to heat exchange ormixing, it is desirable to maintain the flow above RecrII ,because eruption phenomenon intensifies momentum andheat exchange. For other applications, it may be crucialto stay below the RecrII number, because eruptions pro-voke significant perturbations of the pressure and velocityfields.The method which was used in this work was tested thor-oughly by applying it to a variety of simple test problemswith analytical solutions: Poiseuille flow, second Stokesproblem, and well known benchmarks: cavity, backwardstep flow. The results were published in [1]. The rate ofconvergence of the method was shown to be ∼ 1.5. The

obtained results were in agreement with experimental dataand other numerical methods.AcknowledgmentsThe author would like to thank Chia-Luen Lee for hercareful editing in grammar of the article above.Nomenclatured initial distance between the vortex patch centerand the wallH hight of the computational domainh, ∆x, ∆y computational mesh sizeL length of the computational domainN number of vortex particlesRe Reynolds numberr initial radius of the vortex patcht time coordinateu velocity vectoru x component of the velocity vectorv y component of the velocity vectorx, y, α position vectorsx, y space coordinatesΓ circulation (intensity) of the vortex particleγ intensity of the vorticity layerη symmetric cut-off functionν viscosityφ interpolation kernelψ stream functionω vorticity∂t partial time derivative∂x , ∂y partial space derivatives∂2x , ∂2

y second order partial space derivatives

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Figure 7. The considered flow for Re = 10000. The eruptions of the boundary layer and the creation of another near-wall recirculation zone canbe observed for (t = 28, t = 56).

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Figure 8. The considered flow for, Re = 50000. Multiple process of eruptions.

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References

[1] Kudela H., Malecha Z.M., Task Quarterly 13, 15(2008)[2] Batchelor G.K, An introduction to aid dynamics,(Cambridge University Press, 1970)[3] Bradshaw P., Progress in Aerospace Sciences 32, 575(1996)[4] Kudela H., Malecha Z.M., Fluid Dyn. Res. 41(5),055502 (2009)[5] Smith C. R., Walker D. A., In: S.I. Green (Ed.), FluidVortices (Kluwer, 1996) 235[6] Kudela H., Malecha Z.M., Journal of Theoretical andApplied Mechanics 45, 785 (2007)[7] Prandtl L., Verhandlg. III. Intern. Math. Kongr. Hei-delberg, 574 (1905)[8] Schlichting H., Boundary-Layer Theory, (McGraw-Hill Book Company, 1979)

[9] Rott N., Q. Appl. Math. 13(1), 444 (1956)[10] Sears W.R., J. Aeronaut Sci. 23(1), 490 (1956)[11] Telionis D.P., Unsteady viscous ow, (Springer-VerlagNew York Inc., 1981)[12] Cottet G.H., Koumoutsakos P., Vortex method: theoryand practice, (Cambridge University Press, 2000)[13] Wu J. Z., Ma H.Y., Zhou M.D., Vorticity and vortexdynamics, (Springer, 2006)[14] Chorin A.J., J. Fluid Mech. 57, 785 (1973)[15] Koumoutsakos P., Direct Numerical Simulations ofUnsteady Separated Flows Using Vortex Methods,PhD thesis (California Institute of Technology, 1993)[16] Koumoutsakos P., Leonard A., Pepin F., Journal ofComputational Physics 113, 52 (1994)[17] Weinan E., Jian-Guo L., Journal of ComputationalPhysics 124, 368 (1996)[18] Lighthill M.J., Introduction. Boundary Layer Theory,edited by J. Rosenhead (Oxford Univ. Press, 1963)

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Vortex patch with wall interaction as new benchmark test